Abstract
The vertex arboricity va(G) of a graph G is the minimum number of colors the vertices can be colored so that each color class induces a forest. It was known that \(va(G)\le 3\) for every planar graph G. In this paper, we prove that \(va(G)\le 2\) if G is a planar graph without intersecting 5-cycles.
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References
Chartrand G, Kronk HV (1969) The point-arboricity of planar graphs. J Lond Math Soc 44:612–616
Chartrand G, Kronk HV, Wall CE (1968) The point-arboricity of a graph. Israel J Math 6:169–175
Chen M, Raspaud A, Wang WF (2012) Vertex-arboricity of planar graphs without intersecting triangles. Eur J Combin 33:905–923
Choi I, Zhang HH (2014) Vertex arboricity of toroidal graphs with a forbidden cycle. Discrete Math 333:101–105
Fijavž G, Juvan M, Mohar B, Škrekovski R (2002) Planar graphs without cycles of specific lengths. Eur J Combin 23:377–388
Huang DJ, Wang WF (2013) Vertex arboricity of planar graphs without chordal 6-cycles. Int J Comput Math 90:258–272
Huang DJ, Shiu WC, Wang WF (2012) On the vertex-arboricity of planar graphs without 7-cycles. Discrete Math 312:2304–2315
Huang F, Wang XM, Yuan JJ (2014) On the vertex-arboricity of \(K_5\)-minor-free graphs of diameter 2. Discrete Math 322:1–4
Raspaud A, Wang WF (2008) On the vertex-arboricity of planar graphs. Eur J Combin 29:1064–1075
Roychoudhury A, Sur-Kolay S (1995) Efficident algorithms for vertex arboricity of planar graph. Foundations of software technology and theoretical computer science, vol 1026. Springer, Berlin, pp 37–51
Wang WF, Lih K-W (2002) Choosability and edge choosability of planar graphs without five cycles. Appl Math Lett 15:561–565
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This work is supported by NSFC (11271006, 11631014), XJEDU2016I046 and XJEDU2014S067 of China.
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Cai, H., Wu, J. & Sun, L. Vertex arboricity of planar graphs without intersecting 5-cycles. J Comb Optim 35, 365–372 (2018). https://doi.org/10.1007/s10878-017-0168-3
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DOI: https://doi.org/10.1007/s10878-017-0168-3